In a passband communication system the transmitted signal typically undergoes time offset (delay), phase shift and attenuation (amplitude change). These effects must be compensated for at the receiver, and the performance of the receiver can depend greatly on the accuracy of the estimates of these parameters. In the present case, we assume that the time offset has been previously handled and we focus on the problem of estimating the phase shift and attenuation of the signal at the receiver. We consider signalling constellations that have symbols evenly distributed on the complex unit circle, such as, binary phase shift keying (BPSK), quaternary phase shift keying (QPSK) and M−-ary phase shift keying (M-PSK). In this case, the transmitted symbols take the form:si=ejui,  (1)where j=√{square root over (−1)} and ui is from the set
  {      0    ,                  2        ⁢        π            M        ,    …    ⁢                  ,                  2        ⁢                  π          ⁡                      (                          M              -              1                        )                              M        }and M≧2 is the size of the constellation. We assume that some of the transmitted symbols are pilot symbols known to the receiver and the remainder are information carrying data symbols with phase that is unknown to the receiver. So,
                              s          i                ⁢                  {                                                                                          p                    i                                                                                        i                    ∈                    P                                                                                                                    d                    i                                                                                        i                    ∈                    D                                                                        ,                                              (        2        )            where P is the set of indices describing the position of the pilot symbols pi, and D is a set of indices describing the position of the data symbols di. The sets P and D are disjoint, ie P∩D=Ø, and L=|P∪D| is the total number of symbols transmitted.
We assume that time offset estimation has been performed and that L noisy M-PSK symbols are observed (received) by the receiver. The received signal is then,yi=a0si+wi,iεP∪D,  (3)where wi is noise and a0=ρ0ejθ0 is a complex number representing both carrier phase θ0 and amplitude ρ0 (by definition ρ0 is a positive real number). Our aim is to estimate a0 from the noisy symbols {yi,iεP∪D}. Without loss of generality, the L noisy signals may form a block of symbols. The block may be an arbitrary number of symbols selected by the receiver or the size of the block may be determined based upon a communication system parameter such as a predetermined frame size. Complicating matters is that the data symbols {di,iεD} are not known to the receiver and must also be estimated. For the sake of clarity, we define a pilot symbol as a symbol which is known to the receiver and data symbols as symbols which are unknown to the receiver. Thus, data symbols which are known can be treated as pilot symbols in the discussion that follows.
One approach is the least squares estimator, that is, the minimisers of the sum of squares function
                                          SS            ⁡                          (                              a                ,                                  {                                                            d                      i                                        ,                                          i                      ∈                      D                                                        }                                            )                                =                                          ⁢                                                    ∑                                  i                  ∈                                      P                    ⋃                    D                                                              ⁢                              |                                                      y                    i                                    -                                      as                    i                                                  ⁢                                  |                  2                                                      =                                                            ∑                                      i                    ∈                    P                                                  ⁢                                                                  ⁢                                                                                                                        y                        i                                            -                                              as                        i                                                                                                  2                                            +                                                ∑                                      i                    ∈                    D                                                  ⁢                                                                  ⁢                                                                                                                        y                        i                                            -                                              ad                        i                                                                                                  2                                                                    ,                            (        4        )            where |x| denotes the magnitude of the complex number x. The least squares estimator is also the maximum likelihood estimator under the assumption that the noise sequence {wi,iε} is additive white and Gaussian. However, the estimator also works well under less stringent assumptions. The existing literature mostly considers what is called non-coherent detection where no pilot symbols exist (P=Ø where Ø is the empty set). In the non-coherent setting differential encoding is often used and for this reason the estimation problem has been called multiple symbol differential detection. Differential detection comprises determining the difference between the received, phases of two consecutive symbols to determine the encoded phase. That is, the symbol is encoded based upon the change in phase in successive symbols, and thus, unlike the coherent case, the receiver does not need to estimate the carrier phase. A popular approach to multiple symbol differential detection is the so called non-data aided, sometimes also called non-decision directed, estimator based on the paper of Viterbi and Viterbi (A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inform. Theory, vol. 29, no. 4, pp. 543-551, July 1983). The idea is to ‘strip’ the modulation from the received signal by taking yi/|yi| to the power of M. A function F: is chosen, and the estimator of the carrier phase θ0 is taken to be
      1    M    ⁢  ∠  ⁢          ⁢  Awhere ∠ denotes the complex argument and
                    A        =                              1            L                    ⁢                                    ∑                              i                ∈                                  P                  ⋃                  D                                                      ⁢                                          F                ⁡                                  (                                      |                                          y                      i                                        |                                    )                                            ⁢                                                                    (                                                                  y                        i                                                                    |                                                  y                          i                                                |                                                              )                                    M                                .                                                                        (        5        )            
Various choices for F are suggested in the Viterbi and Viterbi paper and a statistical analysis is presented. However, as this paper is only concerned with the non-coherent case, it is not obvious how the pilot symbols should be included in this method.
Thus, there is a need to provide an estimator for the carrier phase and amplitude in a received signal which includes both pilot symbols known to a receiver and data symbols unknown to a receiver, or at least to provide a useful alternative to existing estimation methods.